\section{Math Symbols}

\begin{table}[h!]
\caption{Math Symbols}
\resizebox{\textwidth}{!}
{
\begin{tabular}{|l|l|}
\hline
Name  & Symbol \\
\hline
Vector - bm & $\va\vb\vc\vd\ve\vf\vg\vh\vi\vj\vk\vl\vm\vn\vo\vp\vq\vr\vs\vt\vu\vv\vw\vx\vy\vz$ \\
Element - \$ \$ & $\eva\evb\evc\evd\eve\evf\evg\evh\evi\evj\evk\evl\evm\evn\evo\evp\evq\evr\evs\evt\evu\evv\evw\evx\evy\evz$ \\
\hline
Matrix - bm & $\mA\mB\mC\mD\mE\mF\mG\mH\mI\mJ\mK\mL\mM\mN\mO\mP\mQ\mR\mS\mT\mU\mV\mW\mX\mY\mZ$ \\
Element - \$ \$ & $\emA\emB\emC\emD\emE\emF\emG\emH\emI\emJ\emK\emL\emM\emN\emO\emP\emQ\emR\emS\emT\emU\emV\emW\emX\emY\emZ$ \\
\hline
Random vector - mathbf & $\rva\rvb\rvc\rvd\rve\rvf\rvg\rvh\rvi\rvj\rvk\rvl\rvm\rvn\rvo\rvp\rvq\rvr\rvs\rvt\rvu\rvv\rvw\rvx\rvy\rvz$ \\
Element - textnormal & $\erva\ervb\ervc\ervd\erve\ervf\ervg\ervh\ervi\ervj\ervk\ervl\ervm\ervn\ervo\ervp\ervq\ervr\ervs\ervt\ervu\ervv\ervw\ervx\ervy\ervz$ \\
\hline
Random Matrix - mathbf & $\rmA\rmB\rmC\rmD\rmE\rmF\rmG\rmH\rmI\rmJ\rmK\rmL\rmM\rmN\rmO\rmP\rmQ\rmR\rmS\rmT\rmU\rmV\rmW\rmX\rmY\rmZ$ \\
Element - textnormal & $\ermA\ermB\ermC\ermD\ermE\ermF\ermG\ermH\ermI\ermJ\ermK\ermL\ermM\ermN\ermO\ermP\ermQ\ermR\ermS\ermT\ermU\ermV\ermW\ermX\ermY\ermZ$ \\
\hline
Tensor & $\tA\tB\tC\tD\tE\tF\tG\tH\tI\tJ\tK\tL\tM\tN\tO\tP\tQ\tR\tS\tT\tU\tV\tW\tX\tY\tZ$ \\
Element & $\etA\etB\etC\etD\etE\etF\etG\etH\etI\etJ\etK\etL\etM\etN\etO\etP\etQ\etR\etS\etT\etU\etV\etW\etX\etY\etZ$ \\
\hline
Set - mathbb & $\sA\sB\sC\sD\sE\sF\sG\sH\sI\sJ\sK\sL\sM\sN\sO\sP\sQ\sR\sS\sT\sU\sV\sW\sX\sY\sZ$ \\
Graph - mathcal & $\gA\gB\gC\gD\gE\gF\gG\gH\gI\gJ\gK\gL\gM\gN\gO\gP\gQ\gR\gS\gT\gU\gV\gW\gX\gY\gZ$ \\
\hline
\hline
Base & abcdefghijklmnopqrstuvwxyz \\
Base & ABCDEFGHIJKLMNOPQRSTUVWXYZ \\
textnormal & $\tnla\tnlb\tnlc\tnld\tnle\tnlf\tnlg\tnlh\tnli\tnlj\tnlk\tnll\tnlm\tnln\tnlo\tnlp\tnlq\tnlr\tnls\tnlt\tnlu\tnlv\tnlw\tnlx\tnly\tnlz$ \\
textnormal & $\tnlA\tnlB\tnlC\tnlD\tnlE\tnlF\tnlG\tnlH\tnlI\tnlJ\tnlK\tnlL\tnlM\tnlN\tnlO\tnlP\tnlQ\tnlR\tnlS\tnlT\tnlU\tnlV\tnlW\tnlX\tnlY\tnlZ$ \\
\hline
\$ \$ & $abcdefghijklmnopqrstuvwxyz$ \\
\$ \$ & $ABCDEFGHIJKLMNOPQRSTUVWXYZ$ \\
mathnormal & $\mnla\mnlb\mnlc\mnld\mnle\mnlf\mnlg\mnlh\mnli\mnlj\mnlk\mnll\mnlm\mnln\mnlo\mnlp\mnlq\mnlr\mnls\mnlt\mnlu\mnlv\mnlw\mnlx\mnly\mnlz$ \\
mathnormal & $\mnlA\mnlB\mnlC\mnlD\mnlE\mnlF\mnlG\mnlH\mnlI\mnlJ\mnlK\mnlL\mnlM\mnlN\mnlO\mnlP\mnlQ\mnlR\mnlS\mnlT\mnlU\mnlV\mnlW\mnlX\mnlY\mnlZ$ \\
\hline
\hline
mathbf & $\mbfa\mbfb\mbfc\mbfd\mbfe\mbff\mbfg\mbfh\mbfi\mbfj\mbfk\mbfl\mbfm\mbfn\mbfo\mbfp\mbfq\mbfr\mbfs\mbft\mbfu\mbfv\mbfw\mbfx\mbfy\mbfz$ \\
mathbf & $\mbfA\mbfB\mbfC\mbfD\mbfE\mbfF\mbfG\mbfH\mbfI\mbfJ\mbfK\mbfL\mbfM\mbfN\mbfO\mbfP\mbfQ\mbfR\mbfS\mbfT\mbfU\mbfV\mbfW\mbfX\mbfY\mbfZ$ \\
\hline
mathrm & $\mrma\mrmb\mrmc\mrmd\mrme\mrmf\mrmg\mrmh\mrmi\mrmj\mrmk\mrml\mrmm\mrmn\mrmo\mrmp\mrmq\mrmr\mrms\mrmt\mrmu\mrmv\mrmw\mrmx\mrmy\mrmz$ \\
mathrm & $\mrmA\mrmB\mrmC\mrmD\mrmE\mrmF\mrmG\mrmH\mrmI\mrmJ\mrmK\mrmL\mrmM\mrmN\mrmO\mrmP\mrmQ\mrmR\mrmS\mrmT\mrmU\mrmV\mrmW\mrmX\mrmY\mrmZ$ \\
\hline
mathit & $\mita\mitb\mitc\mitd\mite\mitf\mitg\mith\miti\mitj\mitk\mitl\mitm\mitn\mito\mitp\mitq\mitr\mits\mitt\mitu\mitv\mitw\mitx\mity\mitz$ \\
mathit & $\mitA\mitB\mitC\mitD\mitE\mitF\mitG\mitH\mitI\mitJ\mitK\mitL\mitM\mitN\mitO\mitP\mitQ\mitR\mitS\mitT\mitU\mitV\mitW\mitX\mitY\mitZ$ \\
\hline
mathtt & $\mtta\mttb\mttc\mttd\mtte\mttf\mttg\mtth\mtti\mttj\mttk\mttl\mttm\mttn\mtto\mttp\mttq\mttr\mtts\mttt\mttu\mttv\mttw\mttx\mtty\mttz$ \\
mathtt & $\mttA\mttB\mttC\mttD\mttE\mttF\mttG\mttH\mttI\mttJ\mttK\mttL\mttM\mttN\mttO\mttP\mttQ\mttR\mttS\mttT\mttU\mttV\mttW\mttX\mttY\mttZ$ \\
\hline
mathsf & $\msfa\msfb\msfc\msfd\msfe\msff\msfg\msfh\msfi\msfj\msfk\msfl\msfm\msfn\msfo\msfp\msfq\msfr\msfs\msft\msfu\msfv\msfw\msfx\msfy\msfz$ \\
mathsf & $\msfA\msfB\msfC\msfD\msfE\msfF\msfG\msfH\msfI\msfJ\msfK\msfL\msfM\msfN\msfO\msfP\msfQ\msfR\msfS\msfT\msfU\msfV\msfW\msfX\msfY\msfZ$ \\
\hline
mathfrak & $\mfraka\mfrakb\mfrakc\mfrakd\mfrake\mfrakf\mfrakg\mfrakh\mfraki\mfrakj\mfrakk\mfrakl\mfrakm\mfrakn\mfrako\mfrakp\mfrakq\mfrakr\mfraks\mfrakt\mfraku\mfrakv\mfrakw\mfrakx\mfraky\mfrakz$ \\
mathfrak & $\mfrakA\mfrakB\mfrakC\mfrakD\mfrakE\mfrakF\mfrakG\mfrakH\mfrakI\mfrakJ\mfrakK\mfrakL\mfrakM\mfrakN\mfrakO\mfrakP\mfrakQ\mfrakR\mfrakS\mfrakT\mfrakU\mfrakV\mfrakW\mfrakX\mfrakY\mfrakZ$ \\
\hline
\hline
mathbb & $\mbbA\mbbB\mbbC\mbbD\mbbE\mbbF\mbbG\mbbH\mbbI\mbbJ\mbbK\mbbL\mbbM\mbbN\mbbO\mbbP\mbbQ\mbbR\mbbS\mbbT\mbbU\mbbV\mbbW\mbbX\mbbY\mbbZ$ \\
mathcal &$\mcalA\mcalB\mcalC\mcalD\mcalE\mcalF\mcalG\mcalH\mcalI\mcalJ\mcalK\mcalL\mcalM\mcalN\mcalO\mcalP\mcalQ\mcalR\mcalS\mcalT\mcalU\mcalV\mcalW\mcalX\mcalY\mcalZ$ \\
mathscr & $\mscrA\mscrB\mscrC\mscrD\mscrE\mscrF\mscrG\mscrH\mscrI\mscrJ\mscrK\mscrL\mscrM\mscrN\mscrO\mscrP\mscrQ\mscrR\mscrS\mscrT\mscrU\mscrV\mscrW\mscrX\mscrY\mscrZ$ \\
\hline
\end{tabular}
}
\end{table}

\section{Default Notation}

\centering{\textbf {Numbers and Arrays}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1in}p{3.5in}}
$\displaystyle a$ & A scalar (integer or real) \\
$\displaystyle \va$ & A vector \\
$\displaystyle \mA$ & A matrix \\
$\displaystyle \tA$ & A tensor \\
$\displaystyle \mI_n$ & Identity matrix with $n$ rows and $n$ columns \\
$\displaystyle \mI$ & Identity matrix with dimensionality implied by context \\
$\displaystyle \ve^{(i)}$ & Standard basis vector $[0,\dots,0,1,0,\dots,0]$ with a 1 at position $i$ \\
$\displaystyle \text{diag}(\va)$ & A square, diagonal matrix with diagonal entries given by $\va$ \\
$\displaystyle \ra$ & A scalar random variable \\
$\displaystyle \rva$ & A vector-valued random variable \\
$\displaystyle \rmA$ & A matrix-valued random variable \\
\end{tabular}
\egroup
\vspace{0.5cm}

\centering{\textbf {Sets and Graphs}}
\bgroup
\def\arraystretch{1.5}

\begin{tabular}{p{1.5in}p{3.5in}}
$\displaystyle \sA$ & A set \\
$\displaystyle \sR$ & The set of real numbers \\
$\displaystyle \{0, 1\}$ & The set containing 0 and 1 \\
$\displaystyle \{0, 1, \dots, n \}$ & The set of all integers between $0$ and $n$ \\
$\displaystyle [a, b]$ & The real interval including $a$ and $b$ \\
$\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$ \\
$\displaystyle \sA \backslash \sB$ & Set subtraction, i.e., the set containing the elements of $\sA$ that are not in $\sB$ \\
$\displaystyle \gG$ & A graph \\
$\displaystyle \parents_\gG(\ervx_i)$ & The parents of $\ervx_i$ in $\gG$
\end{tabular}
\vspace{0.5cm}

\centering{\textbf {Indexing}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1.5in}p{3.5in}}
$\displaystyle \eva_i$ & Element $i$ of vector $\va$, with indexing starting at 1 \\
$\displaystyle \eva_{-i}$ & All elements of vector $\va$ except for element $i$ \\
$\displaystyle \emA_{i,j}$ & Element $i, j$ of matrix $\mA$ \\
$\displaystyle \mA_{i, :}$ & Row $i$ of matrix $\mA$ \\
$\displaystyle \mA_{:, i}$ & Column $i$ of matrix $\mA$ \\
$\displaystyle \etA_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor $\tA$ \\
$\displaystyle \tA_{:, :, i}$ & 2-D slice of a 3-D tensor\\
$\displaystyle \erva_i$ & Element $i$ of the random vector $\rva$ \\
\end{tabular}
\egroup
\vspace{0.5cm}

\centering{\textbf {Calculus}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1.5in}p{3.5in}}
% NOTE: the [2ex] on the next line adds extra height to that row of the table.
% Without that command, the fraction on the first line is too tall and collides
% with the fraction on the second line.
$\displaystyle\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$ \\ [2ex]
$\displaystyle \frac{\partial y} {\partial x} $ & Partial derivative of $y$ with respect to $x$ \\
$\displaystyle \nabla_\vx y $ & Gradient of $y$ with respect to $\vx$ \\
$\displaystyle \nabla_\mX y $ & Matrix derivatives of $y$ with respect to $\mX$ \\
$\displaystyle \nabla_\tX y $ & Tensor containing derivatives of $y$ with respect to $\tX$ \\
$\displaystyle \frac{\partial f}{\partial \vx} $ & Jacobian matrix $\mJ \in \R^{m\times n}$ of $f: \R^n \rightarrow \R^m$ \\
$\displaystyle \nabla_\vx^2 f(\vx)\text{ or }\mH( f)(\vx)$ & The Hessian matrix of $f$ at input point $\vx$\\
$\displaystyle \int f(\vx) d\vx $ & Definite integral over the entire domain of $\vx$ \\
$\displaystyle \int_\sS f(\vx) d\vx$ & Definite integral with respect to $\vx$ over the set $\sS$ \\
\end{tabular}
\egroup
\vspace{0.5cm}

\centering{\textbf {Probability and Information Theory}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1.5in}p{3.5in}}
$\displaystyle P(\ra)$ & A probability distribution over a discrete variable \\
$\displaystyle p(\ra)$ & A probability distribution over a continuous variable, or over a variable whose type has not been specified \\
$\displaystyle \ra \sim P$ & Random variable $\ra$ has distribution $P$ \\% so thing on left of \sim should always be a random variable, with name beginning with \r
$\displaystyle  \E_{\rx\sim P} [ f(x) ]\text{ or } \E f(x)$ & Expectation of $f(x)$ with respect to $P(\rx)$ \\
$\displaystyle \Var(f(x)) $ &  Variance of $f(x)$ under $P(\rx)$ \\
$\displaystyle \Cov(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P(\rx)$ \\
$\displaystyle H(\rx) $ & Shannon entropy of the random variable $\rx$ \\
$\displaystyle \KL ( P \Vert Q ) $ & Kullback-Leibler divergence of P and Q \\
$\displaystyle \mathcal{N} ( \vx ; \vmu , \mSigma)$ & Gaussian distribution % over $\vx$ with mean $\vmu$ and covariance $\mSigma$ \\
\end{tabular}
\egroup
\vspace{0.5cm}

\centering{\textbf {Functions}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1.5in}p{3.5in}}
$\displaystyle f: \sA \rightarrow \sB$ & The function $f$ with domain $\sA$ and range $\sB$ \\
$\displaystyle f \circ g $ & Composition of the functions $f$ and $g$ \\
  $\displaystyle f(\vx ; \vtheta) $ & A function of $\vx$ parametrized by $\vtheta$.
  (Sometimes we write $f(\vx)$ and omit the argument $\vtheta$ to lighten notation) \\
$\displaystyle \log x$ & Natural logarithm of $x$ \\
$\displaystyle \sigma(x)$ & Logistic sigmoid, $\displaystyle \frac{1} {1 + \exp(-x)}$ \\
$\displaystyle \zeta(x)$ & Softplus, $\log(1 + \exp(x))$ \\
$\displaystyle || \vx ||_p $ & $\normLp$ norm of $\vx$ \\
$\displaystyle || \vx || $ & $\normLtwo$ norm of $\vx$ \\
$\displaystyle x^+$ & Positive part of $x$, i.e., $\max(0,x)$ \\
$\displaystyle \bm{1}_\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise \\
\end{tabular}
\egroup
\vspace{0.5cm}

\centering{\textbf {Machine Learning}}
\bgroup
\def\arraystretch{1.5}
\begin{tabular}{p{1.5in}p{3.5in}}
$\train$ & is \\
$\valid$ & is \\
$\test$ & is \\
$\pdata$ & is \\
$\ptrain$ & The empirical distribution defined by the training set \\
$\Ptrain$ & The empirical distribution defined by the training set \\
$\pmodel$ & The model distribution \\
$\Pmodel$ & The model distribution \\
$\ptildemodel$ & The model distribution \\
$\pencode$ & Stochastic autoencoder distributions \\
$\pdecode$ & Stochastic autoencoder distributions \\
$\precons$ & Stochastic autoencoder distributions \\
\end{tabular}
\egroup
\vspace{0.5cm}
